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Section What is a Numerical Solution?
Most of what weβve done, so far, has been focused on finding
exact solutions to differential equations, like
\(y(t) = e^{-3t} \sin(2t)\text{.}\) This kind of solution is called an
analytic solution , or sometimes a
closed-form solution . It is valuable because it expresses
\(y(t)\) as a formula-like structure that you can plug in any value of
\(t\) and instantly get the exact
\(y\) -value.
However, many differential equations, such as
\begin{equation*}
y' = e^{-y^2}, \quad y(0) = 0.
\end{equation*}
simply donβt have a tidy closed-form solution. In those cases, we switch tools. Instead of searching for an exact formula, we use a
numerical method . A numerical method doesnβt hand you
\(y(t)\) as a formulaβit builds an
approximation , one step at a time, starting from what you know and using the differential equation to predict what happens next. The result is a
numerical solution .
Checkpoint 111 . πβ Analytic Solution, AKA β¦.
What is another name for an analytic solution?
Closed-form solution
Yes , a closed-form solution is another name for an analytic solution.
Approximate solution
No , an approximate solution refers to a numerical solution.
Continuous solution
Not all analytic solutions are continuous , and not all continuous solutions are analytic.
Systematic solution
While you may solve analytically using a systematic process, this is not a name for an analytic solution.
Subsection Analytic vs. Numerical Solutions
To see the difference between analytic and numerical solutions, consider the initial value problem:
\begin{equation}
y' = y, \quad y(0) = 1, \quad 0 \le t \le 2\tag{25}
\end{equation}
Analytically, this equation has the elegant solution:
\begin{equation*}
y(t) = e^t\text{.}
\end{equation*}
With this formula, you can calculate the exact value at any time. For example, \(t = 0.65\) gives
\begin{equation*}
y(0.65) = e^{0.65} \approx 1.915540829...\text{.}
\end{equation*}
The same equation can also be solved numerically. Instead of a formula, we produce a table of
\(t\) -values and approximate
\(y\) -values:
\(t\)
\(y(t)\) (approx)
0.00
1.0000
0.25
1.2500
0.50
1.5625
0.75
1.9531
β¦
β¦
Plot the
points
\(\rightarrow\)
Checkpoint 112 . πβ Recognizing a Numerical Solution.
When you say youβve found a βnumerical solutionβ to an initial value problem, what do you actually have?
A list of values approximating \(y(t)\) at specific times
Correctβnumerical solutions are tables of approximate values, not formulas.
A formula expressing \(y(t)\) using functions like \(\sin t\) or \(e^t\)
No, that describes an analytic solution.
An exact solution valid for all \(t\)
No, numerical solutions are approximations.
A graph showing all possible solutions
No, that describes a slope field, not a numerical solution.
The analytic solution gives a smooth curve for every
\(t\text{.}\) The numerical solution gives
dots βa sequence of approximations. Connect those dots and you get a picture of the solutionβs shape, even though no formula was found.
Table 114. Analytic vs. Numerical Solution Summary
\(\textbf{Analytic Solutions}\)
\(\textbf{Numerical Solutions}\)
\(\textbf{Also Known As}\)
\(y = \) formula in
\(t\)
Table of
\(t\) &
\(y\) values
\(\textbf{Solution Values}\)
\(\textbf{Solution Graph}\)
Subsection Why Do We Need Numerical Methods?
At first glance, analytic solutions might seem βbetterβ than numerical ones. But there are important reasons why numerical methods arenβt just usefulβtheyβre essential:
Many equations simply donβt have a closed-form solution.
Even when they do, the formula might be so complicated itβs impractical.
Numerical methods are easier to tweak if the model or data changes.
Checkpoint 115 . πβ Why use numerical methods?
Which of the following are good reasons to use a numerical method?
They can approximate a solution when no closed-form solution exists.
Trueβsometimes theyβre the only option.
They can be automated on a computer.
Trueβtheyβre ideal for computer simulations.
They are easier to adjust when the equation changes.
Trueβnumerical methods adapt easily to model tweaks.
They produce more accurate solutions than analytic methods.
Noβanalytic solutions are exact when they exist.
π€ Wrap-Up.
ποΈ Key Takeaways...
Analytic solutions give formulas; numerical solutions give a list of values.
Analytic solutions are exact; numerical solutions are approximate.
Numerical solutions are essential when analytic solutions are too complicated or donβt exist.
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